General sextic polynomial root formulae via Tschirnhausen transformations and inverse regularized beta functions

The general sextic equation, when transformed into a reduced form with three free coefficients, possesses known closed-form solutions for certain specific cases involving symmetries between coefficients and cancellation of coefficients. The general case wherein there are no assumed symmetries or cancellations remains unknown, as well as some reduced cases. This work seeks to complete the theory of solving the general sextic equation with real coefficients by collecting known results into one paper and adding some novel methods to cover the unsolved general case and thus reconstruct some unsolved degenerate cases. The methods center on the regularized beta function with parameters four and three, which evaluates to a reduced sextic equation. Introducing three free parameters into a formulation of the regularized beta function and then equating coefficients with the three-parameter reduced sextic allows for explicit formulations of its six roots via the multi-valued inverse regularized beta function. Discussions of degenerate cases involve biquadratic and bicubic equations as well as one Bring-Jerrard quintic equation. Each case includes coefficient criteria for real and non-real roots as well as multiplicity of roots.